Background: The inverse problem solution in the field of ElectroEncephaloGraphy (EEG) analysis has been addressed in the scientific literature for many decades, utilizing either mathematical techniques for measurement fitting or pure ElectroMagnetic (EM) methods involving complex head models for the prediction of the near field.
New method: A novel radiated EM field estimation analysis scheme is proposed for EEG analysis, based on the determination of a grid of equivalent distributed EM sources with equal magnetic moments, in order to compute the extrapolated far field. A Pattern Search approach is adopted to minimize the Mean Absolute Relative Error between the EM near field created by the source grid and the EM field extracted by the measurements.
Results: The application of the method on a subject’s brain activity recordings in the context of “Protagoras” mental-auditory experiment demonstrates the capability of the proposed scheme to compare the subject’s concentration differences between the limit of present and past versus the limit of present and future.
Comparison with existing methods: The proposed method combines features from different existing methods, both in terms of mathematical and EM theory techniques, in order to extend their capabilities and transform the conventional analysis of EEG recordings to a far field radiation basis.
Conclusions: The treatment of the brain as an equivalent far field radiator can be a useful and promising new perspective to the established analysis of EEG recordings arising from brain activity during mental processing.
The prediction of a set of Electro-Magnetic (EM) sources that are able to reproduce measured ElectroEncephaloGraphy (EEG) and/or Event-Related Potential (ERP) skull surface potentials at a set of measurement points has attracted the attention of the scientific community over the last years (Grech et al., 2008; Safavi et al., 2018; Song et al., 2015; Safavi et al., 2016; Ala et al., 2015a; Hanna-Leena and Lauri, 2018; Wei et al., 2018; Ala et al., 2017, 2015b). In order to solve this challenging problem, various mathematic techniques have been studied throughout the literature. The aforementioned inverse problem exhibits high complexity and uncertainty, due to the different subjects’ skull dimensions and the EM properties of the human skull and brain (Gabriel et al., 1996; Hoekema et al., 2003; Sadleir and Argibay, 2007). In order to overcome this obstacle, the scientists employ classic EM field theory in order to formulate the skull surface current density (Hjorth,1991). This approach has the significant advantage of employing a wellknown and well established mathematical background. Apart from the inherent complexity of the inverse problem, the task of determining the real sources of brain activity (neurons) is difficult, due to the extremely small number of electrodes used to measure the brain activity, compared
to the numerous neurons needed to be determined (Pakkenberg and Gundersen, 1997; West and Gundersen, 1990). Consequently, scientists aim at reduced-variables’ inverse problem methods (Grech et al., 2008; Safavi et al., 2018, 2016), appropriate for the extraction of behavioral statistical conclusions of the study of interest. These techniques employ various algorithms such as Minimum Norm Estimates (MNE), Weighted Minimum Norm Estimates (WMNE), Low-Resolution Electrical Tomography (LORETA), Quadratic regularization and spatial regularization (S-MAP), Local AUtoRegressive Average (LAURA), non-linear Least-Squares problem, Beamforming, Brain Electric Source Analysis (BESA), Multiple Signal Classification algorithm (MUSIC), Simulated Annealing (SA), Genetic Algorithms (GAs) and Artificial Neural Networks (ANN).
In this paper, a novel method for the inverse problem solution is presented. The proposed method utilizes a mesh of distributed EM sources with equal magnetic moment. Based on the functional description of sets of brain neurons forming closed current paths (Purves et al., 2001), magnetic dipoles are chosen to be used as elementary sources, placed in parallel to the measurements’ surface and allowing the prediction of both the near and the far electric field produced by the actual subjects’ brain activity. The parameters of the magnetic dipoles are determined by the surface potential measurements extracted by the “Protagoras” test campaign described below to characterize/analyze the human brain activity in the context of the relevant experiments.
The aforementioned test campaign aims at shedding light to the cognitive representation of the limits (boundaries) of the concept present in relation to both its past and future. To achieve this, the theories of philosopher John Mc Taggart, introduced at the beginning of the last century are employed, namely the A and B theory of time are employed. According to the A-theory, which sometimes is also called tensed theory, all events are ordered in terms of temporal properties like being past, being present and being future. According to the B-theory, which is called tenseless theory, all events are ordered in terms of temporal relations like earlier than, simultaneous with and later than (McTaggart, 1908; Craig and Craig, 2000, 2010; Ingthorsson, 2016).
The tensed theory holds that time is a dynamic aspect of existence. Time flows and objects change in time. The grammatical tenses used in everyday language, was, is, will be, have an ontological correspondence. The structure of time is mirrored by the structure of grammar. That concept was first introduced by the famous sophist Protagoras (490–420 BCE). He was the first to distinguish the tenses of the verb and to emphasize the power of the right moment (Dillon, 2003).
Thus, the “Protagoras” experiment was designed to demonstrate that the tenses are coherent by explaining exactly how they differ from each other. In particular, the present study was designed to contrast two mental functions: delta EEG activity associated with the mental processing of the limit (boundary) between present and past vs the limit (boundary) between present and future. This was performed while the participants were exposed to tensed tasks engaging Working Memory (WM) operation. At this point it is useful to mention that WM is believed to be a system for temporarily storing and managing the information required to carry out complex cognitive operations such as reasoning (Baddeley, 2012; Hasson et al., 2015).
The rest of the paper is organized as follows: In Section 2, a short description of the measurement data on which the proposed method will be applied, the performed pre-processing techniques, the mathematical formulation of the relevant theory and the description of the proposed method are presented. Furthermore, in Section 3 the application of the proposed method over the measurement data is presented, followed by the results discussion in Section 4. Finally, in Section 5 the
results of the method are summarized, and the possible expansions of the method are outlined.
Materials and methods
- “Protagoras” test campaign
As already mentioned, the “Protagoras” experiment was designed to compare mental processing between present and past vs the limit (boundary) between present and future utilizing auditory stimuli. Participants were exposed to three arrays of 30 tensed verbs classified as ‘positive’, ‘negative’, and ‘neutral’ and 15 existential tenses deriving from the verbs “be”, “have” and “make”. Presentation order was pseudorandom and counterbalanced across participants. The test procedure has been carried out in accordance with The Code of Ethics of the World Medical Association (Declaration of Helsinki) and the subjects gave their consent for experimentation. The measurements were conducted in an electromagnetically shielded room minimizing interference caused by external EM fields and an LISN (Line Impedance Stabilization Network) has been used to minimize possible conducted emissions. Each participant was instructed through the intercom to hear carefully each verb’s tense uttering through both earphones and according to the frequency of the followed warning stimulus to pay his or her attention on the limit (boundary) between present and past if the stimulus was high (3 kHz), or on the limit (boundary) between present and future if the warning stimulus was low (500 Hz). The ERP recordings have 1 s duration for each stimulus, while the EEG recordings have 2 s duration for each stimulus. The sampling period of the measured data is 1 ms and consequently the sampling frequency is 1 kHz. The method outline follows the next scheme:
- The recording starts.
- The first 400 msec are occupied from the pronunciation of a specific verb from the test pool. The selected verb, regardless of the stimulus that follows, is pronounced in its three tenses, namely past, present and future.
- Next, pause 1 lasts for 500ms.
- Next, a warning stimulus of 100 msec in duration is heard. The frequency of the warning stimulus is 500 Hz for the subject to concentrate on the limit between past and present tense or 3 kHz for the subject to concentrate on the limit between present and future tense of the relevant verb.
- Next, pause 2 lasting 2900 msec follows. This is the main recording part of the test campaign, where the data for ERP and EEG measurements exist. In this period, the subject focuses on the appropriate limit, based on the warning stimulus that heard before.
- Finally, a second warning stimulus of the same frequency with the previous one that lasts for 100 msec is heard and declares the end of the specific verb test recording.
- A pause of 4 up to 9 s follows, until the start of the next verb test. In this period, the subject declares how well was concentrated on the given stimulus. This period is not recorded.
43 volunteers were tested throughout the test campaign and their EEG and ERPs data from 32 electrodes are available for conventional analysis (α, β, γ, δ, θ bands) as well as the P50, N100, N200, P200, P300 and P600 waveforms.
- Method outline
Prior to the description of the method, the following assumptions need to be considered. At first, the EM properties of the subjects’ brains materials are not taken into account. This consideration is based on similar computational EM problems, such as the Method of Auxiliary Sources (MAS) (Papakanellos and Capsalis, 2003) as consequently the proposed method does not aim on the identification of the extremely large number of real sources (neurons) that produce the measured surface potentials, but instead focuses on the determination of a finite number of equivalent elementary EM sources that would produce the electric field calculated by the measured surface potentials in free space. As a result of this principle, the solutions to the inverse problem are utilized in order to extract an equivalent EM model of the human brain, exhibiting the capability of extrapolating the emitted electric field both in the near and the far region (Spantideas et al., 2016). Thus, the initial set of measured surface potentials is transformed into a set of equivalent EM models, offering a new perspective to the existing EEG and ERP analysis methods.
For this reason, the proposed method utilizes a swarm of gridded equivalent elementary EM sources with relatively low-power emissions, aiming at a distributed source formation, rather than a lumped source network. Furthermore, in this paper, infinitesimal magnetic dipoles (closed current loops) are used as equivalent elementary sources, because of their EM functional similarity with the neurons’ closed circuits.
The meaning of the obtained results might be better understood considering the psychophysiological implication of delta EEG activity. The origin of delta waves during cognitive processes remains unknown. It has been hypothesized that low frequency oscillations of delta and theta ranges are associated with motivational and emotional processes (Knyazev, 2007). The role of delta waves during mental tasks has also been postulated to be associated with cortical deafferentation or with the inhibition of the sensory afferences that interfere with internal concentration (Harmony, 2013). The fact that delta activity is characteristic of a state in which interneurons and the thalamocortical inputs are inactive strongly suggests that delta oscillations’ far-field radiation power increase during mental tasks inhibits all the interferences that may affect the performance of the task. Another possible mechanism in relation to inhibition for delta oscillations has been proposed in (Knyazev and Slobodskaya, 2003). They speculated that delta oscillations are linked with the most ancient phylogenetic system of information transmission and that behavioral inhibition is associated with a stronger mechanism of descending inhibition, in which higher systems inhibit the lower. This mechanism was measured by negative correlations between delta, theta and alpha powers.
Thus, the observed increase of delta EEG activity accompanying the allocation of attention on the limit (boundary) between present and future as compared to that between present and past, suggest to fit with the second law of thermodynamic. The second law of thermodynamics with its fundamental property directing towards dissipation (entropy) fits with our unquestionable sense of time flowing from past to present and finally to future. Indeed, the far-field power increase of delta frequencies during mental tasks is associated with functional cortical deafferentation, or inhibition of the sensory afferences that interfere with internal concentration. These inhibitory oscillations would modulate
the activity of those networks that should be inactive to accomplish the task (Di Lazzaro et al., 2009).
Changes in this frequency band appear to be suitable to provide additional and complimentary evidence to our understanding regarding the time experience. The sensitivity of this analysis demonstrated by these findings indicates also that using this method in EEG studies is promising to shed new light on current theories of the time perception and increase our understanding of their underlying neural processes.
In this paper, a novel inverse problem method for existing EEG and fields of the corresponding predicted magnetic MDMs indicative to the far-field radiated power through Eq. (11) are displayed in Table 1 both
for the ‘past’ and the ‘future’ stimuli. The Appendix section Table A1 contains the description of the Verb IDs utilized in the method application. The verbs are presented in their original (Greek) form, as well as in English translation. Student’s t-test value for these two samples is 2.81, which is significantly higher than the corresponding Student’s distribution p-value at 0.95 significance value, which is F-1 ν(0.95) = 1.68, (v denotes the degrees-of-freedom) indicating that the expected mean value of the ‘past neutral’ verbs cluster is clearly higher than the ‘future neutral’ verbs cluster. The respective t-test values for the other participants are between 1.81 and 2.81, strengthening the previous conclusion. Moreover, this remark is also verified through the examination of P300 (ERP), where it has been found that for this cluster of verbs the mean temporal energy for all the electrodes in the future tense is higher than the one in the past tense. These results, on the one hand verify the proposed method validity and on the other hand lead to the extraction of an important conclusion: the concentration towards the ‘future’ results in higher brain activation than the concentration towards the ‘past’, regarding the ‘neutral’ verbs. This finding will be compared in future work with the results from other subjects, in order to extract valid general conclusions about this issue.